The convergence of Euler products
The author's point of departure is Riemann's hypothesis that the classical zeta-function \(\zeta(s)\) has no zeros in the half-plane \(\text{Re }s>1/2\). He says: `The present work supports the conjecture that the Dirichlet zeta-function \(\zeta_ \chi(s)\) has no zeros in the half-plane \(\text{Re }s>1/2\) when \(\chi\) is a non-principal character modulo \(r\). The results do not apply without modification to the principal character.' His method is to associate certain Hilbert spaces of entire functions with \(\zeta_ \chi(s)\) using an interpretation of the Euler product for \(\zeta_ \chi(s)\). Some of these Hilbert spaces, which he calls Sonine spaces, rely upon properties of the Hankel transformation. The author obtains recurrence relations, differential equations, and asymptotic behaviour for the defining functions, but the results are too technical to state here. Modified versions of his results are derived for the classical zeta-function \(\zeta(s)\) with principal character modulo 1. They extend his earlier results such as [Bull. Am. Math. Soc., New Ser. 15, 1-17 (1986; Zbl 0602.10028)].
scientific article; zbMATH DE number 90039
scientific article